3.388 \(\int \frac{1}{x^{7/2} \sqrt{b x^2+c x^4}} \, dx\)

Optimal. Leaf size=149 \[ \frac{5 c^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right ),\frac{1}{2}\right )}{21 b^{9/4} \sqrt{b x^2+c x^4}}+\frac{10 c \sqrt{b x^2+c x^4}}{21 b^2 x^{5/2}}-\frac{2 \sqrt{b x^2+c x^4}}{7 b x^{9/2}} \]

[Out]

(-2*Sqrt[b*x^2 + c*x^4])/(7*b*x^(9/2)) + (10*c*Sqrt[b*x^2 + c*x^4])/(21*b^2*x^(5/2)) + (5*c^(7/4)*x*(Sqrt[b] +
Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(21
*b^(9/4)*Sqrt[b*x^2 + c*x^4])

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Rubi [A]  time = 0.181635, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2025, 2032, 329, 220} \[ \frac{5 c^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{21 b^{9/4} \sqrt{b x^2+c x^4}}+\frac{10 c \sqrt{b x^2+c x^4}}{21 b^2 x^{5/2}}-\frac{2 \sqrt{b x^2+c x^4}}{7 b x^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(7/2)*Sqrt[b*x^2 + c*x^4]),x]

[Out]

(-2*Sqrt[b*x^2 + c*x^4])/(7*b*x^(9/2)) + (10*c*Sqrt[b*x^2 + c*x^4])/(21*b^2*x^(5/2)) + (5*c^(7/4)*x*(Sqrt[b] +
Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(21
*b^(9/4)*Sqrt[b*x^2 + c*x^4])

Rule 2025

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(j - 1)*(c*x)^(m - j +
1)*(a*x^j + b*x^n)^(p + 1))/(a*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IntegerQ[p] && LtQ[0, j,
n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[m + j*p + 1, 0]

Rule 2032

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(c^IntPart[m]*(c*x)^FracP
art[m]*(a*x^j + b*x^n)^FracPart[p])/(x^(FracPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]), Int[x^(m
+ j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && PosQ[n
- j]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rubi steps

\begin{align*} \int \frac{1}{x^{7/2} \sqrt{b x^2+c x^4}} \, dx &=-\frac{2 \sqrt{b x^2+c x^4}}{7 b x^{9/2}}-\frac{(5 c) \int \frac{1}{x^{3/2} \sqrt{b x^2+c x^4}} \, dx}{7 b}\\ &=-\frac{2 \sqrt{b x^2+c x^4}}{7 b x^{9/2}}+\frac{10 c \sqrt{b x^2+c x^4}}{21 b^2 x^{5/2}}+\frac{\left (5 c^2\right ) \int \frac{\sqrt{x}}{\sqrt{b x^2+c x^4}} \, dx}{21 b^2}\\ &=-\frac{2 \sqrt{b x^2+c x^4}}{7 b x^{9/2}}+\frac{10 c \sqrt{b x^2+c x^4}}{21 b^2 x^{5/2}}+\frac{\left (5 c^2 x \sqrt{b+c x^2}\right ) \int \frac{1}{\sqrt{x} \sqrt{b+c x^2}} \, dx}{21 b^2 \sqrt{b x^2+c x^4}}\\ &=-\frac{2 \sqrt{b x^2+c x^4}}{7 b x^{9/2}}+\frac{10 c \sqrt{b x^2+c x^4}}{21 b^2 x^{5/2}}+\frac{\left (10 c^2 x \sqrt{b+c x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+c x^4}} \, dx,x,\sqrt{x}\right )}{21 b^2 \sqrt{b x^2+c x^4}}\\ &=-\frac{2 \sqrt{b x^2+c x^4}}{7 b x^{9/2}}+\frac{10 c \sqrt{b x^2+c x^4}}{21 b^2 x^{5/2}}+\frac{5 c^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{21 b^{9/4} \sqrt{b x^2+c x^4}}\\ \end{align*}

Mathematica [C]  time = 0.0141234, size = 57, normalized size = 0.38 \[ -\frac{2 \sqrt{\frac{c x^2}{b}+1} \, _2F_1\left (-\frac{7}{4},\frac{1}{2};-\frac{3}{4};-\frac{c x^2}{b}\right )}{7 x^{5/2} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(7/2)*Sqrt[b*x^2 + c*x^4]),x]

[Out]

(-2*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[-7/4, 1/2, -3/4, -((c*x^2)/b)])/(7*x^(5/2)*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.184, size = 134, normalized size = 0.9 \begin{align*}{\frac{1}{21\,{b}^{2}} \left ( 5\,\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}{x}^{3}c+10\,{c}^{2}{x}^{4}+4\,bc{x}^{2}-6\,{b}^{2} \right ){x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(7/2)/(c*x^4+b*x^2)^(1/2),x)

[Out]

1/21/(c*x^4+b*x^2)^(1/2)/x^(5/2)*(5*((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2)*2^(1/2)*((-c*x+(-b*c)^(1/2))/(-b*c
)^(1/2))^(1/2)*(-x*c/(-b*c)^(1/2))^(1/2)*EllipticF(((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2))*(-b*c)
^(1/2)*x^3*c+10*c^2*x^4+4*b*c*x^2-6*b^2)/b^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c x^{4} + b x^{2}} x^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(c*x^4+b*x^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(c*x^4 + b*x^2)*x^(7/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{x}}{c x^{8} + b x^{6}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(c*x^4+b*x^2)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*x^4 + b*x^2)*sqrt(x)/(c*x^8 + b*x^6), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{\frac{7}{2}} \sqrt{x^{2} \left (b + c x^{2}\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(7/2)/(c*x**4+b*x**2)**(1/2),x)

[Out]

Integral(1/(x**(7/2)*sqrt(x**2*(b + c*x**2))), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c x^{4} + b x^{2}} x^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(c*x^4+b*x^2)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(c*x^4 + b*x^2)*x^(7/2)), x)